Reconstruction of a two-dimensional reflecting medium over a circular domain: Exact solution

If a two-dimensional acoustic reflectivity function is defined within the interior of a circle, reflectivity data may be acquired by transmitting acoustic pulses from isotropic elements distributed around the circumference of the circle and recording the resulting backscattered sound as a function of time. If the reflectivity function fulfills the conditions of an ’’idealized’’ weakly reflecting medium, the resulting pulse–echo data may be regarded as the line integrals of this function defined over circular arcs centered at points lying on the circumference of the enclosing circle. In this paper we show that on the basis of these line integrals the unknown reflectivity in the interior of the circle can be expressed explicitly in terms of its line integrals defined over the set of paths consisting of all circular arcs whose centers lie on the circumference of the enclosing circle. We propose that the resulting reconstruction equations could provide the foundation for a new approach to reflectivity tomography. A numerical example is also given.